A geometric proof of Jung’s Theorem
by Stéphane Lamy
This is a translation of my paper “Une preuve géométrique du théorème de Jung”, Enseign. Math. 48 (2002), no. 3-4, 291–315.
1
Introduction
The complex affine space Cn presents all the qualities of the fascinating mathematical
objects: very basic in nature, it stands as the starting point of a vast number of interesting
and difficult problems. In particular the group Aut[Cn ] of polynomial automorphisms of
Cn is far for being well understood. The study of these automorphisms is of course closely
related to the research around the famous Jacobian Conjecture (see [12]). But many more
questions about the group Aut[Cn ] are also natural : we would like to determine the
finite subgroups, the Lie subgroups, the linearizable subgroups... One can find in [21] a
beautiful survey of these problematics. Furthermore it recently became clear that these
automorphisms provide some example of dynamical systems with a very rich behavior. A
pioneer work in this direction is [13]; for a panorama of the progresses made along the last
ten years one can read [35]. All these questions are delicate in general, not to mention
the possibility to study what happens when we replace C by an arbitrary field, or even
by a ring. Nevertheless there exists a particular case for which we have a lot of results :
the dimension 2 case. Indeed there exists a structure theorem, stated by H.W.E. Jung as
early as 1942, that gives a set of generators for Aut[C2 ].
We will denote by A the group of affine automorphisms of C2 , i.e. the group of
elements of Aut[C2 ] that extend to holomorphic automorphisms of P2 ; and we will name
E (for “elementary”, following the notations of [13]) the subgroup of Aut[C2 ] composed
of the automorphisms that preserve the pencil of lines y = constant. In other words:
A = {(x, y) 7→ (a1 x + b1 y + c1 , a2 x + b2 y + c2 ); ai , bi , ci ∈ C, a1 b2 − a2 b1 6= 0};
E = {(x, y) 7→ (αx + P (y), βy + γ); α, β ∈ C∗ , γ ∈ C, P ∈ C[X]}.
Theorem 1 (Jung, 1942). The group Aut[C2 ] of polynomial automorphisms of C2 is
generated by the affine and elementary automorphisms.
A few years after Jung, this result is made more precise by W. Van der Kulk as follows:
Theorem 2 (Van der Kulk, 1953). Given a field k (of any characteristic, possibly not
algebraically closed), the group of polynomial automorphisms of k 2 is generated by the
affine et elementary automorphisms with coefficients in k. Furthermore Aut[k 2 ] is the
amalgamated product of these two subgroups.
After the articles of Jung [19] and Van der Kulk [22] many other proofs, using different
technics, have been proposed. The purpose of this paper being to give one another proof,
we begin, in order to explain our motivations, by giving a quick survey of the available
1
proofs in the literature. Generally speaking, the common idea of all this proofs is to
proceed by induction on the degree; thus given an automorphism
g : (x, y) 7→ (g1 (x, y), g2 (x, y))
where g1 , g2 are some polynomials of respective degrees d1 and d2 , the point is to prove
that it is possible to lower the degree of g by successive compositions by an affine and an
elementary automorphisms. More precisely, composing by an affine automorphism we can
suppose that d1 is strictly less than d2 ; then it remains to prove that the homogeneous
component of higher degree of g1 is a multiple of the one of g2 , which is quite easily
deduced from the fact that d1 is a multiple of d2 .
The proof most similar to ours (that is to say, geometric in nature) is maybe the one by
M. Nagata [28], who is inspired by the article of Van der Kulk. Previously, W. Engel [11]
had proposed a proof, which is took on by A. Gutwirth [17]. Nagata laconically comments
on these two proofs by saying that they seem difficult to read for him. Anyway the idea
here is to extend g to a birational map of P2 and to consider the curve C preimage by g
of a general line. We then obtain some informations on the degrees d1 and d2 by studying
the singularity of C at infinity.
With the aim to give a proof on an arbitrary field, L. Makar-Limanov [23] proposed
an alternative, completely algebraic approach to the proof of Van der Kulk. The idea is to
introduce a new degree by allowing different weights to the variables x and y, according
to the degrees associated with the inverse map of g. Note that few years later this same
author propose by a similar approach a description of the automorphism group of a large
class of affine surfaces [24]. A proof proposed by W. Dicks in 1983 [9] is a simplified version
of the argument of Makar-Limanov; one can find a precise redaction of this proof in the
book of P. M. Cohn [8].
A slightly different approach has been proposed by R. Rentschler. It is quite easy once
the Jung-Van der Kulk’s theorem is established to show that any algebraic representation
of (C, +) in Aut[C2 ] is given up to conjugation by elementary automorphisms. Rentschler
takes the converse way : he proposes to show that property first, and then remarks that we
can deduce the Jung’s theorem from it. Indeed, to the automorphism g we can associate
the locally nilpotent derivation ∂/∂g1 . This proof, published as a note to the CRAS in
1968 [31], has been rewritten in details recently by L. M. Drużkowski and J. Gurycz [10].
A so-called elementary proof is published in 1988 by J. H. McKay and S. S. Wang
[26]; it is based on an inversion formula. The authors show that the application g −1
can be expressed with the help of resultant computations which involve the one variable
polynomials g1 (0, t), g1 (t, 0), g2 (0, t), g2 (t, 0). The expected relation between d1 and d2
follows.
On the other hand, one can find an “sophisticated” proof in the book of K. Matsuki
[25]. The idea here is to use the framework of Mori theory to formulate a proof of the
Jung’s theorem, with the hope that this can lead to a breakthrough in the study of the
group Aut[Cn ] for n ≥ 3.
Let us cite finally one more approach : in [1] S. S. Abhyankar and T. T. Moh prove
that two biregular embeddings of C into C2 are the same up to an automorphism of C2 ,
and remark that their proof implies the Jung’s theorem. A few authors had proposed
some new proofs of this result; one can cite the recent articles of R. V. Gurjar [16], E.
Casas-Alvero [6] and E. Artal-Bartolo [3], the three of them proposing some geometric
proofs. Nevertheless we will see that if the only aim is to get a proof of the Jung’s result
then it is possible to give a much more concise geometric proof.
The starting point of our work is a very natural postulate: the Jung’s theorem is
a result of birational geometry. Indeed any automorphism of C2 can be extended as a
birational map of P2 . In this context it seems to us that to proceed by an induction on the
2
degree is not the more natural reasoning; the number of indeterminacy point turns out to
be a more easily managed quantity. This was not really the point of view of Jung, in spite
of his title : “On the birational entire transformations of the plane”. Nevertheless, in a note
that seems to have been forgotten, O.-H. Keller [20] reacts to the work of Jung noticing,
without any details, that it should certainly be possible to give a simplified proof using
the basic theory of birational maps of P2 . Later, in a short article, I.R. Shafarevich [32]
states the Jung’s theorem and indicates that the demonstration is based on the possibility
to write down any birational map between surfaces as a sequence of blow-ups (this is
Theorem 6 stated in the next paragraph); unfortunately he does not seem to have ever
got the opportunity to publish such a proof (in the complement [34] to his first article
Shafarevich merely refers to a paper by M.H. Gizatullin and V.I. Danilov [14] which by its
ambition of maximal generality turns out to be quite difficult to read). Finally, recently S.
Orevkov [30] hints that we can recover the Jung’s theorem from a work of A. G. Vitushkin,
but again the details are not made explicit.
We can indeed draw a parallel between the Jung’s theorem and a classical result
generally attributed to M. Noether [29]:
Theorem 3 (Noether, 1872). Any birational map of the projective plane P2 admits a
decomposition with the help of linear automorphisms and of the standard quadratic involution
σ : [x : y : z] 99K [yz : xz : xy].
It seems that the first complete proof of this statement is in fact due to G. Castelnuovo
[7], who deduces the Noether’s theorem from the following intermediate result :
Theorem 4 (Castelnuovo, 1901). Any birational map of the projective plane P2 can be
written as a composition of linear automorphisms and of so-called de Jonquières maps.
Concerning de Jonquières maps let us simply mention that they are the maps of degree
n which admit a base point of multiplicity n − 1; the noteworthy fact is that the polynomial automorphisms that extend as de Jonquières maps are precisely (up to conjugation
by an affine automorphism) the elementary automorphisms. Thus the Jung’s theorem
may be seen as a special case of the result of Castelnuovo. One can find in [27] a proof of
Theorem 3 very closed in spirit to the proof of the Jung’s theorem we give in this article.
We may wonder that the theorem of Castelnuovo goes back to 1901, whereas those of
Jung which turns out to be an easier special case (in particular it will not be necessary
to use the notion of multiplicity of an indeterminacy point as Nagata does) goes back to
1942. A possible answer is that neither Castelnuovo nor any of his contemporaries has ever
thought about this problem. One can summarize our reasoning saying that we propose
ourselves to give the proof of the Jung’s theorem as a geometer from the beginning of
the twentieth century could have conceived it; or in other words, the proof that seems
to be hidden behind the remarks of Keller and Shafarevich cited above. Our proof is
concise, does not need any computation, and explains why this result is very particular to
the case of the dimension 2. The method being geometric in nature, it has seemed more
transparent to us to remain in the classical setting (that is, we work on the field C); nevertheless this restriction is by no mean essential as we will remark at the end of the article.
The paper is organized as follows.
The second paragraph gathers the results of birational geometry that we use; these are
very elementary and you can find them in your favorite introductory book to algebraic
geometry (which is probably [15], [18] or [33]).
The proof of Jung’s theorem is detailled in the third paragraph.
Finally, in the last paragraph we illustrate our method with an example and then we
proceed to prove the theorem of Van der Kulk. First we indicate how to show geometrically
3
that Aut[C2 ] is the amalgamated product of his affine and elementary subgroups. We
should note that this is really a trivial remark (which certainly turns out to be crucial
in practice), and that the really delicate result is the one in the Jung’s statement. In
conclusion, we show how our proof can be easily adapted to the case of an arbitrary field.
2
Birational maps between surfaces
Our reasoning to prove the Jung’s theorem is to consider a polynomial automorphism of
C2 as a birational map from P2 to itself, and then to use a classical structure theorem for
these applications. Among many possible choices we have chosen as a reference for this
section the first two chapters of [4].
By a surface we always mean a smooth complex projective surface, and an open set
is always a Zariski open set. Let X and Y be two surfaces; a rational map ϕ : X 99K Y
is a morphism from an open set U in X to Y , that can not be extended to a larger open
set. When U = X we have a true morphism : we reserve the notation ϕ : X → Y to this
case. It is easy to show (see [4, II.4]) that X \ U is a finite set. Thus strictly speaking a
rational map is not a map, because there exists a finite number of points where ϕ is not
well defined. Nevertheless the image of a curve is always well defined : if C is a curve in
X, we define the strict transform ϕ(C) of C by ϕ as the adherence of the image by ϕ of
C ∩ U . Note that the image of a (say, irreducible) curve could be a point.
A birational map between X and Y is a rational map ϕ : X 99K Y that induces an
isomorphism between an open set of X and an open set of Y .
Example. Consider the following map from P2 to itself (that we already met in the
statement of Noether’s theorem):
σ : [x : y : z] 99K [yz : xz : xy].
The map σ, called standard quadratic map, is well defined except at the three points
[1 : 0 : 0], [0 : 1 : 0] and [0 : 0 : 1]. Furthermore σ induces an automorphism of P2 minus
the three lines x = 0, y = 0 and z = 0. We leave to the reader to check for example that
the image by σ of the line z = 0 is the point [0 : 0 : 1], that the image of a line through
[0 : 0 : 1] is still a line through [0 : 0 : 1], and that the image of a general line is a conic
passing through the three points [1 : 0 : 0], [0 : 1 : 0] and [0 : 0 : 1].
A fundamental example of birational map is the blow-up of a point, that we recall
briefly. Let S be a surface, and p a point in S. There exists a surface S̃ and a morphism
π : S̃ → S such that
• E = π −1 (p) is isomorphic to P1 ;
• π induces an isomorphism from S̃ \ E to S \ p.
Up to isomorphism S̃ and π are unique. As a matter of terminology, we say that π
is the blow-up map at the point p, or that S̃ is the blow-up of S at p; the rational curve
E is called the exceptional divisor of the blow-up. If C ⊂ S is a curve through p, we
denote by C̃ the strict transform of C, which is the adherence of π −1 (C \ {p}). By the
total transform of C we mean the divisor π ∗ C; for instance if C is smooth in p we have
π ∗ C = C̃ + E.
The surface S is endowed with an intersection form : if D1 , D2 are two divisors (i.e.
P
some finite sums
λi Ci where the Ci are irreducible, possibly singular, curves, and the
λi are relative integers), then it is possible to define an intersection number D1 .D2 . When
D1 and D2 are simply two distinct irreducible curves, D1 .D2 is the number of intersection
4
points of these two curves, counted with multiplicities; D1 .D2 is in this case a positive
number. We can extend this natural definition to make sense of the intersection number
of two arbitrary divisors, in particular we can speak of the self-intersection of a divisor
(see [4, th. I.4]). We note D2 instead of D.D the self-intersection of a divisor D. Note
that the self-intersection of a curve could be negative. The intersection number satisfies
the following agreeable properties (D1 , D2 and D3 are three divisors):
• If D2 and D3 are linearly equivalent then D1 .D2 = D1 .D3 ;
• With the above notations :
(π ∗ D1 .π ∗ D2 ) = (D1 .D2 );
(E.π ∗ D1 ) = 0.
Regarding the influence of a blow-up on the intersection numbers, we will use repetitively
the following equalities that are easily deduced from the properties just stated (C is always
a smooth curve through p):
Formulas 5.
E 2 = −1;
C̃ 2 = C 2 − 1.
Let us make precise a point of vocabulary. According to how we consider the map
S̃ → S we will use two different terminologies : we will say that one goes from S to S̃
by the blow-up of the point p, and that one goes from S̃ to S by the contraction of the
curve E. In the sequel we will consider some sequences of blow-ups. If we note πpi the
blow-up map at the point pi , we will have some maps of the form ϕ : M 7→ X, where M
and X are surfaces and ϕ = πpn ◦ · · · ◦ πp1 (here p1 ∈ X and for all i ≥ 2, pi belongs to the
surface obtained after the blow-ups of the points p1 , · · · , pi−1 ). In this situation we will
say that p1 is the first point blown-up by ϕ, or conversely that the exceptional divisor En
produced by πpn is the first curve contracted by ϕ.
The blow-ups are sufficient to describe any birational maps between surfaces : this is
what the following result states precisely (see [4, II.12]).
Theorem 6 (Zariski, 1944). Any birational map between two surfaces is obtained as a
sequence of blow-ups followed by a sequence of contractions ; in other words if X, Y are
some surfaces and
g : X 99K Y
is a birational map (which is not an isomorphism), then there exists a surface M and
some sequences of blow-ups π1 and π2 such that the following diagram commutes:
MA
|| AAA π2
|
AA
|
||
AA
|~ |
X _ _ _ _g _ _ _/ Y
π1
Following Beauville we attribute this theorem to Zariski. The proof, which is not too
difficult, is made in two steps. In the first step we compose g with a sequence of blow-ups
π1 in order to get rid of the indeterminacy points. Thus we obtain a commutative diagram:
MA
|| AAA ḡ
|
AA
|
||
AA
|~ |
_
_
_
_
_
_
_/ Y
X
g
π1
5
where ḡ is a morphism. Note that this process can be applied to any rational map between
surfaces (see [4, II.7]); and there exists a similar statement in higher dimension.
On the other hand the second step, which consists to show that the morphism ḡ is a
sequence of contractions (see [4, II.11]) is very special to the case of a birational morphism
between two surfaces. The heart of the proof is the following proposition:
Proposition 7. Let ḡ : M 7→ Y be a birational morphism between surfaces. If y ∈ Y is a
point where ḡ −1 is not well-defined, then ḡ admits a decomposition
> Ỹ ?
~~ ??? σ
~
??
~
??
~~
~~
/Y
M
h
ḡ
where σ is the blow-up mat at y, and h is a morphism.
We propose a proof of this proposition using an elementary argument from differential
geometry, that may throw light on the proof given in [4, II.8]. Note that we do not use at
any moment the Castelnuovo criterion (contraction of rational curves with self-intersection
−1). We admit the
Lemma 8 (see [4, II.10]). If ϕ : X 99K Y is a birational map between two surfaces, and
if x ∈ X is a point where ϕ is not well defined, then there exists a curve C ⊂ Y such that
ϕ−1 (C) = x.
Proof of Proposition 7. Suppose that h = σ −1 ◦ ḡ is not a morphism, and let x ∈ M be
a point where h is not well defined. In this situation : on one hand ḡ(x) = y and ḡ is
not locally injective at x; on the other hand there exists a curve in Ỹ that is contracted
on x by h−1 . This curve must be the exceptional divisor E associated with σ. Let p and
q be two distinct points on E where h−1 is well defined, and let C, C 0 be two germs of
smooth curves transversal to E in p and q respectively. Then σ(C) and σ(C 0 ) are two
transverse germs of smooth curves in y, which are image by ḡ of two germs of curves in x.
The differential of ḡ in x is then of rank 2, which contradicts the fact that ḡ is not locally
injective in x (see figure 1).
Ỹ
•
C0
C
q
p
•
h
@
@σ
@
TT
Z
@
Z
σ(C 0 )
Y
M
h−1 (C 0 )
E
•x
ḡ
X
X
y
•
h−1 (C)
σ(C)
Figure 1: h non well defined in x leads to a contradiction.
Before we start with the proof of Jung’s theorem we would like to make precise some
points of terminology and explain in which context we will use the Zariski’s theorem. We
6
will call indeterminacy points of g the points we have to blow-up in the construction of
π1 ; thus these are either points on X or on a surface obtained by blow-ups from X. The
indeterminacy points contained in X are said to be proper (classically one says that the
other points are in some infinitely near neighborhood of the proper indeterminacy points).
The number of indeterminacy points of g (proper or not) will be noted #ind(g).
Remark. Note that this definition is consistent because the sequences of blow-ups π1
and π2 produced by the theorem are uniquely determined by g (up to isomorphism). The
sequence π1 is made by blowing-up exactly the points where g is not well-defined. Similarly
the sequence π2 is entirely determined by the points where g −1 is not well defined. Of
course we could artificially make the sequences π1 and π2 longer by blowing-up some points
where g and g −1 are well defined. However it was implicit in our statement of Zariski’s
theorem that we consider some minimal sequences π1 and π2 , in the sense that we have
the following universal property (see [2]):
Let ϕ1 : M 0 7→ X and ϕ2 : M 0 7→ Y be two birational morphisms such that ϕ2 = g ◦ϕ1 .
Then there exists a unique morphism h : M 0 7→ M such that the following diagram
commute :
M10
1
h 111
111 ϕ
ϕ1 2
{M BB 111
BB 1
{{
B 1
{{{π1
π2 BBB11
{
}{
!
X _ _ _ _ _ _ _/ Y
g
In the sequel we will use Zariski’s theorem only in a very particular case: we will
consider g : X 99K P2 coming from a polynomial automorphism of C2 . By this we mean
that we have a partition X = C2 ∪ D where D is an union of irreducible curves (called
the divisor at infinity), and a partition P2 = C2 ∪ L where L is a line (line at infinity),
such that g induces an isomorphism from X \ D to P2 \ L. This situation implies some
strong restrictions on the indeterminacy points of g; this is what the following lemma
makes precise:
Lemma 9. Let X be a surface and g be a birational map from X to P2 coming from a
polynomial automorphism of C2 . We suppose that g is not a morphism. Then
1. g admits a unique proper indeterminacy point, located on the divisor at infinity of
X;
2. g admits some indeterminacy points p1 , · · · , ps (s ≥ 1) such that
(a) p1 is the unique proper indeterminacy point;
(b) for all i = 2, · · · , s, the point pi is located on the divisor produced by the blow-up
of pi−1 ;
3. Every irreducible curve contained in the divisor at infinity of X is contracted to a
point by g;
4. the first contracted curve of π2 is the strict transform of a curve contained in the
divisor at infinity of X;
5. in particular, if X = P2 , the first contracted curve by π2 is the strict transform of
the line at infinity in X.
7
Proof. We know (Lemma 8) that if p is a proper indeterminacy point of g then there
exists a curve contracted to p by g −1 . In our situation the only curve of P2 candidate to
be contracted is the line at infinity; so there is at most one proper indeterminacy point for g
in X. As we suppose that g is not a morphism, g admits exactly one proper indeterminacy
point. The second assertion then comes from a straightforward induction. Furthermore,
each curve in the divisor at infinity in X either is contracted to a point, or is sent onto
the line at infinity in P2 . Since g −1 contracts the line at infinity to a point, this latter
possibility is excluded : we have shown the third assertion. From the argument above we
see that the divisor at infinity in M is composed from the divisor with self-intersection
−1 produced by the blow-up of ps , from the other divisors produced by the sequence of
blow-ups, all of which with self-intersection less or equal to −2, and finally from the strict
transform of the divisor at infinity in X (here we have used the formulas 5). The first curve
contracted by π2 must have self-intersection −1, and can not be the last curve produced
by π1 (this would contradict the fact that ps is an indeterminacy point), thus we see that
the first curve contracted by π2 is the strict transform of a curve contained in the divisor
at infinity in X. The last assertion is just another formulation of the fourth one, in the
case X = P2 .
3
Proof of Jung’s theorem
We consider g a polynomial automorphism of C2 , that we extend as a birational map (still
denoted g) of P2 into itself. If g is written
g : (x, y) 7→ (g1 (x, y), g2 (x, y))
and if n is the degree of g (i.e. the maximum of the degrees of g1 and g2 ), then the
extension of g to P2 is written in homogeneous coordinates as
g : [x : y : z] 99K [z n g1 (x/z, y/z) : z n g2 (x/z, y/z) : z n ].
The line at infinity in P2 is the line of equation z = 0. We want to prove that g is a composition of affine and elementary automorphisms. The proof will proceed by induction on
the number #ind(g) of indeterminacy points of g.
By Lemma 9 (assertion 1) the extension g : P2 99K P2 admits a unique indeterminacy
point located on the line at infinity. Composing g by an affine automorphism we can
assume that this point is [1 : 0 : 0]. In other words we have a commutative diagram:
2
P
}> A A
}
g
a }}
A0
}
}
A
}}
P2 _ _ _ _g _ _ _/ P2
where a is affine and g0 admits [1 : 0 : 0] as indeterminacy point. Obviously we have
#ind(g0 ) = #ind(g).
We are now going to prove that there exists a diagram
2
P
}> A A
ϕ }
g ◦ϕ−1
A0
}
A
}
P2 _ _ _ g_0 _ _ _/ P2
8
where ϕ is the extension of an elementary automorphism of C2 , and such that
#ind(g0 ◦ ϕ−1 ) < #ind(g0 ).
Our reasoning will be to consider the diagram given par Zariski’s theorem1
MB
|| BBB π2
|
BB
|
BB
||
~||
2
_
_
_
_
_
_
_/ P2
P
g0
π1
and to reorganize the blow-ups occurring in the sequences π1 and π2 . Thus, in four steps
that we will now detail, ϕ will be constructed from some of the blow-ups of the sequence
π1 and some of the contractions of the sequence π2 .
First step : blow-up of [1 : 0 : 0].
The point [1 : 0 : 0] is the first point blown-up in π1 ; so let us consider the surface
F1 obtained by blowing-up P2 at the point [1 : 0 : 0]. This surface is a completion of C2
which is naturally endowed with a rational fibration coming from the lines y = constant.
The divisor at infinity is composed of two rational curves (i.e. isomorphic to P1 ) meeting
transversally in one point. On one hand we have the strict transform of the line at infinity
in P2 ; this is a fiber that we will denote f∞ . On the other hand we have the exceptional
divisor of the blow-up, which is a section for the fibration : it will be denoted s∞ . Of
2 = 0 and s2 = −1. More generally for all n ≥ 1 we
course (apply formulas 5) we have f∞
∞
will denote by Fn a completion of C2 with a structure of a rational fibration, such that
the divisor at infinity is composed of two transverse rational curves : one fiber f∞ and
one section s∞ with self-intersection −n. These surfaces are classically called Hirzebruch
surfaces; we do not suppose the reader has any particular knowledge of these surfaces. One
point about notation : we will write s∞ (Fn ) and f∞ (Fn ) when more than one Hirzebruch
surface will come into play.
Now we come back to the map g0 . We have a commutative diagram:
F1 A
}>
A g1
}
A
}
A
}
2
_
_
_
_
_
_
_
/ P2
P
g0
(D1)
ϕ1
where ϕ−1
1 is the blow-up map at the point [1 : 0 : 0]. we have
#ind(g1 ) = #ind(g0 ) − 1.
We consider again the diagram obtained from Zariski’s theorem applied to g0 . From
Lemma 9 (assertion 5) the first contracted curve in π2 , which must be a curve with selfintersection −1 in M , is the strict transform of the line at infinity. This is the fiber f∞ in
2 = 0. The self-intersection of this curve still has to drop by one,
F1 . Now in F1 we have f∞
thus the indeterminacy point p of g1 is located on f∞ . Furthermore we know (Lemma 9,
assertion 2) that this same point p belongs to the curve s∞ produced by the blow-up ϕ−1
1 .
We conclude that p is precisely the intersection point of f∞ and s∞ .
Second step : rising induction.
1
Each time we will use Zariski’s theorem we will note M , π1 and π2 the surfaces et the sequences of
blow-ups produced, the context should allow to avoid any confusion.
9
In the following reasoning we use some maps between ruled surfaces generally called
“elementary transformations” (nevertheless we will not use this terminology in order to
avoid any confusion with the elements of the group E). These transformations are defined
as the composition of one blow-up and one contraction. More precisely let S be a ruled
surface, i.e. a surface equipped with a fibration f : S → C where C is a curve, and
such that the fibers of f are all isomorphic to P1 . Take p ∈ S et denote by F the fiber
containing p. The elementary transformation at the point p is the birational map obtained
as the composition of the blow-up of p (producing an exceptional divisor F 0 ) and of the
contraction of the strict transform of F . Thus we obtain a new ruled surface S 0 .
F0
A
A
A
A
A
F
F
S
p
•
blow-up p
Q
Q
Q
Q
f
Q
Q
Q
S0
contract F
C
F0
•
0
f
In the proof of Lemmas 10 and 11 we will use such transformations, applied to ruled
surfaces with basis C isomorphic to P1 .
Lemma 10. Let n ≥ 1, and let h be a birational map from Fn to P2 coming from a
polynomial automorphism of C2 . Suppose that the unique proper indeterminacy point of
h is the intersection point p of f∞ (Fn ) and s∞ (Fn ). Consider the commutative diagram
Fn+1D
z<
D h0
z
D
z
D!
z
Fn _ _ _ _ _ _ _ _/ P2
ϕ
h
where ϕ is the blow-up of p followed by the contraction of the strict transform of f∞ . Then
the birational map h0 = h ◦ ϕ−1 satisfies the following two properties:
1. #ind(h0 ) = #ind(h) − 1;
2. the proper indeterminacy point of h0 is located on f∞ (Fn+1 ).
Proof. We consider the decomposition of h as a sequence of blow-ups:
MA
}} AAA π2
}
AA
}
AA
}}
}~ }
Fn _ _ _ _ _ _ _/ P2
π1
h
The (strict) transform of s∞ (Fn ) in M has self-intersection less or equal to −2; then
Lemma 9 (assertion 4) allows us to conclude that the first contracted curve in π2 is the
transform of f∞ (Fn ). Thus the transform of f∞ (Fn ) in M has self-intersection −1; on the
other hand in Fn we have f∞ (Fn )2 = 0. It follows that after the blow-up of p the rest of
the sequence of blow-ups π1 is performed on points outside of f∞ . Instead of doing these
blow-ups before contracting the transform of f∞ (Fn ) we can reverse the order, that is we
10
can first contract f∞ (Fn ) and then realize the rest of the blow-up sequence. In other words
we have a commutative diagram (πp is the blow-up at p and contf∞ is the contraction of
the transform of f∞ (Fn )):
M
{ ???contf∞
{
?
{
{{
{
{
M0
{{
z 999
{
z
{
99
zz
}{{
99
zz
B
z
πp BBcontf∞ zz
99
B
z
|z
0
h
Fn+1 _ _ _ _ _ _ _ _r/8 P2
Fn
OR
l o
TW Z
\ _ b d g j
h
These two maps, blow-up of p and then contraction of f∞ (Fn ), are summed-up in the following picture where we represent only the divisors at infinity, with their self-intersections
(the latter are computed with the help of Formulas 5). We observe in particular that the
resulting surface is of type Fn+1 .
0
−1
p
•
f∞ (Fn )
f∞ (Fn )
blow-up p
−n
0
e
•
e −1
f∞ (Fn+1 ) = F 0
e
F0 e
e
contract f∞ (Fn )
−(n + 1)
e
−(n + 1)
s∞ (Fn )
s∞ (Fn )
s∞ (Fn+1 )
= s∞ (Fn )
In conclusion, the blow-up p drops by one the number of indeterminacy points, and the
contraction of f∞ (Fn ) do not create a new one : we have #ind(h0 ) = #ind(h) − 1.
Furthermore the indeterminacy point of h0 is located on the curve produced blowing-up
p, that is f∞ (Fn+1 ).
At the end of the first step we are under the conditions of application of Lemma 10,
with n = 1. The lemma then gives a map h0 : F2 99K P2 whose proper indeterminacy
point is located on the fiber f∞ (F2 ). If this point is precisely the intersection point with
the section at infinity, we can apply again the lemma. Iterating this process as long as we
remain under the hypotheses of Lemma 10, we obtain a diagram
Fn A
}>
A g2
}
A
}
A
}
F1 _ _ _ _ _ _ _/ P2
(D2)
ϕ2
g1
where ϕ2 is obtained by applying n − 1 times Lemma 10. Furthermore we have
#ind(g2 ) = #ind(g1 ) − n + 1.
Finally the indeterminacy point of g2 is located on f∞ (Fn ), and is not the intersection
point with s∞ (Fn ) (otherwise we could apply the lemma once more).
Third step : descending induction.
We are going to apply the following lemma, which is similar to Lemma 10 (except that
now we suppose n ≥ 2).
11
Lemma 11. Let n ≥ 2, and let h be a birational map of Fn to P2 that comes from a
polynomial automorphisms of C2 . Suppose that the unique proper indeterminacy point
p of h is located on f∞ , but is not the intersection point of f∞ and s∞ . Consider the
commutative diagram
Fn−1
z<
D
D h0
D
z
D!
z
Fn _ _ _ _ _ _ _ _/ P2
ϕ
z
h
where ϕ is the blow-up of p followed by the contraction of the strict transform of f∞ (Fn ).
Then the birational map h0 satisfies the following two properties:
1. #ind(h0 ) = #ind(h) − 1;
2. the proper indeterminacy point of h0 is located on f∞ (Fn−1 ) and is not the intersection point of f∞ (Fn−1 ) and s∞ (Fn−1 ).
Proof. Consider the decomposition of h given by Zariski’s theorem :
MA
}} AAA π2
}
AA
}
AA
}}
~}}
_
_
_
_
_
_
_ / P2
Fn
π1
h
The transform of s∞ (Fn ) in M has self-intersection −n, since n ≥ 2 we deduce (lemma 9)
that the first contracted curve of π2 is the transform of f∞ (Fn ). As in the proof of lemma
10 we obtain a commutative diagram:
πp
Fn
O
M?
{{ ???contf∞
{
{
{{
{
M 09
{{
z
99
{
z
{
99
zz
z
}{{
99
BB contf∞ zzz
99
BB
z
|zz
0
h
Fn−1 _ _ _ _ _ _ _ _r/8 P2
o
R T
W Z \ _ b d g j l
h
The surface obtained by blowing-up p and contracting the transform of f∞ is of type Fn−1 ;
this is summed up in the following picture.
(non proper) indeterminancy point of h
corresponding to the proper indeterminancy point of h0
F0
p
•
•
0
−1
•
0
f∞ (Fn−1 ) = F 0
f∞ (Fn )
blow-up p
−n
−1
f∞ (Fn )
contract f∞ (Fn )
−n
s∞ (Fn )
s∞ (Fn )
•
−(n − 1)
s∞ (Fn−1 )
= s∞ (Fn )
The equality #ind(h0 ) = #ind(h) − 1 is straightforward. Denoting by F 0 the divisor
produced by the blow-up of p, h admits a (non proper) indeterminacy point located on
F 0 . Furthermore this point can not be the intersection point of F 0 and of the transform
12
of f∞ (Fn ), because otherwise we would have π1−1 (f∞ (Fn )) with self-intersection less or
equal to −2 and this contradicts that it should be the first contracted curve of π2 . Finally
this point is the proper indeterminacy point of h0 , is located on f∞ (Fn−1 ) and is not the
intersection point of f∞ (Fn−1 ) and s∞ (Fn−1 ).
After the second step we are under the hypotheses of Lemma 11. Furthermore if n ≥ 3
then the map h0 produced by the lemma still satisfies the hypotheses of this same lemma.
After applying n − 1 times Lemma 11 we obtain a diagram
F1 @
}>
@ g3
}
@
}
@
}
Fn _ _ _ _ _ _ _/ P2
(D3)
ϕ3
g2
with
#ind(g3 ) = #ind(g2 ) − n + 1.
Finally, the proper indeterminacy point of g3 is located on f∞ (F1 ), and is not the intersection point of f∞ (F1 ) and s∞ (F1 ).
Fourth step : last contraction.
Applying Zariski’s theorem to g3 we get a diagram :
MA
}} AAA π2
}
AA
}
AA
}}
~}}
_
_
_
_
_
_
_/ P2
F1
g3
π1
Lemma 9 (assertion 4) ensures that the first contracted curve in π2 is the strict transform
by π1 of either f∞ or s∞ . Suppose this is the transform of f∞ . Then after the sequence of
blow-ups π1 and the contraction of this curve, the transform of s∞ has self-intersection 0
and thus will not be contracted in the sequel of π2 ; this contradicts the third assertion of
Lemma 9. So the first contracted curve must be the transform of s∞ , and we can contract
the latter straight away to obtain the following diagram:
2
P
~> @ @
ϕ4 ~~~
g
@4
~~
@
~
~
_
_
_
_
_
_
_
/ P2
F1
(D4)
g3
The morphism ϕ4 is the blow-up map with exceptional divisor s∞ , and since it is defined
up to isomorphism we can decide that the point onto which we contract is [1 : 0 : 0].
Furthermore we have
#ind(g3 ) = #ind(g4 ).
Conclusion.
We can add up the four diagrams (D1), · · · , (D4) into only one
13
_ ϕ_2 _/ Fn _ ϕ_3 _/ F1
AAA ϕ4
ϕ1 }
0
AA
AA
}
0
}
0 g1 g 2
2
P PP
2 g3
nP
0
n
PP
n
0 P
n gn4
g0 P P 0 n
P( 2 vn n
F1
}> 0
P
or in more compact form :
2
P
}> A A
ϕ4 ◦ϕ3 ◦ϕ2 ◦ϕ1 }
g
A4
}
A
}
2
_
_
_
_
_
_
_
/ P2
P
g0
with
#ind(g4 ) = #ind(g0 ) − 2n + 1 (where n ≥ 2).
We now just have to check that ϕ = ϕ4 ◦ ϕ3 ◦ ϕ2 ◦ ϕ1 is an elementary automorphism. This
is equivalent to show that ϕ preserve the foliation y = constant, that is that ϕ preserves
the pencil of lines passing through [1 : 0 : 0]. Now this fact is clear : the blow-up ϕ1 sends
the lines through [1 : 0 : 0] to the fibers of F1 , ϕ2 and ϕ3 preserve the fibrations of F1
and Fn , and finally the contraction ϕ4 sends the fibers of F1 to the lines passing through
[1 : 0 : 0]. Thus the map g4 is an automorphism of C2 obtained from g by composing first
with an affine automorphism and then with an elementary automorphism, and we have
the inequality:
#ind(g4 ) < #ind(g).
By induction on #ind(g), this ends the demonstration.
4
4.1
Complements
An example
Let us consider the following automorphism g :
g : (x, y) 7→ (y + β(y + αx2 )2 + γ(y + αx2 )3 , y + αx2 ) with α, β, γ ∈ C∗ .
The decomposition of g is made of two elementary automorphisms :
g(x, y) = (x + βy 2 + γy 3 , y) ◦ (y, x) ◦ (x + αy 2 , y) ◦ (y, x).
Applying Zariski’s theorem to g we decompose g with the help of eight blow-ups and eight
contractions. When we apply our algorithm to g we change the order of these blow-ups
14
and contractions as follows :
MB
||
}||
π1
F1
||
}||
<<<
||
}||
F2
w
ww
{ww
w
ww
{ww
<<<
F1
;;
;
2
_
_
_
_
_
_
_
_
_
_
_
_
/ P2
P S T
2 ,y)◦(y,x)
T(x+αy
U V
W X
Y Y Z
[ \
|
||
|} |
BB
BB
!
A
}} AAA
~}}
BB
BB
!
BB
BB
!
<<<
BBπ2
BB
!
BB
BB
!
<<<
BB
BB
!
<<<
F1
F2
F3
F2
F1 ;
;;
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _5/ P2
k
(x+βy 2 +γy 3 ,y)◦(y,x)
j j
h i
g
e f f
] ] ^ _ ` a a b c d
g
We should explain more precisely how we obtain this diagram. The proper indeterminacy
point of g is [0 : 1 : 0], so we start by considering g ◦ (y, x) which is not well-defined at the
point [1 : 0 : 0]. We blow-up this point, and we apply Lemma 10 once. On the resulting
surface F2 the indeterminacy point is located on a general point of the fiber f∞ (the exact
location depends on the coefficient α). Then we apply Lemma 11 and we contract the
section s∞ (F1 ). Then we have the decomposition :
g = g 0 ◦ (x + αy 2 , y) ◦ (y, x)
where g 0 admits only 5 indeterminacy points. Again we consider g 0 ◦ (y, x) to get an
automorphism with proper indeterminacy point equal to [1 : 0 : 0]. We blow-up this point,
and we can apply twice the lemma 10. Then we can apply Lemma 11 twice (the exact
location of the indeterminacy points on f∞ (F3 ) and f∞ (F2 ) depends on the coefficients γ
and β). Finally we contract the section s∞ (F1 ), and we get
g 0 (x, y) = (x + βy 2 + γy 3 , y) ◦ (y, x).
4.2
Amalgamated product structure
We have shown that Aut[C2 ] is generated by the subgroups A and E, now we want to
prove that Aut[C2 ] is the amalgamated product of these two subgroups. In other words
we want to show that any relations in the group Aut[C2 ] is induced by the relations in the
groups A and E. This is equivalent to show that a composition
h = a1 ◦ e1 ◦ · · · ◦ an ◦ en with ai ∈ A \ E, ei ∈ E \ A
can never be equal to the identity. Note that we can restrict ourselves to such compositions
h, that is of even length and beginning by an affine automorphism. Indeed if h is of odd
length (and ≥ 3 : if h is of length 1 there is nothing to do) we can reduce the length of h
by a suitable conjugation. Furthermore if h is of even length and begins by an elementary
automorphism, we just consider h−1 instead.
Now it is easy to check that each automorphism ei , view as a birational map on
2
P , contracts the line at infinity to the point [1 : 0 : 0] (because we suppose ei 6∈ A).
Furthermore, ai 6∈ E is equivalent to say that the point [1 : 0 : 0] is not a fixed point of ai .
We deduce easily that the extension of h to P2 contracts the line at infinity to the point
a1 ([1 : 0 : 0]), which contradicts that h is distinct from identity.
15
4.3
Proof on an arbitrary field
Given a field k we note Ak and Ek the affine and elementary groups with coefficients in
k; by k̄ we will denote the algebraic cloture of k. A first remark is that our proof works
without any modification in the case of an algebraically closed field k̄ (the characteristic
does not matter). The results on the geometry of surfaces that we use, that is the properties
of the intersection form (formulas 5) and the decomposition theorem of Zariski are stated
with such level of generality for instance in Chapter V of [18]. Similarly we can copy word
to word the above argument to show that Aut[k̄ 2 ] is the amalgamated product of Ak̄ and
Ek̄ .
Consider now a non algebraically closed field k, and let g be an element of Aut[k 2 ] with
degree d. We already know that g is a composition of affine et elementary automorphisms
with coefficients in k̄. We have to show that there exists such a decomposition with only
elements of Ak and Ek .
As above we consider g as a birational map from P2k̄ to itself. The crucial point is
that we know that g admits a unique proper indeterminacy point, which is the image by
g −1 of the line at infinity. Let us choose a point p on the line at infinity all of whose
homogeneous coordinates are in k and which is not the indeterminacy point of g −1 (one
can choose for instance one of the two points [1 : 0 : 0] or [0 : 1 : 0]). Then g −1 (p) is the
proper indeterminacy point of g, and is therefore contained in P2k . A similar reasoning
shows that the proper indeterminacy point of g −1 is also in P2k . Composing g on the right
and on the left by well-chosen elements of Ak we can then suppose that the indeterminacy
points of g and g −1 are both equal to [1 : 0 : 0]. That is to say we are in the case where
the decomposition of g in the amalgamated product of Ak̄ and Ek̄ begins and ends with
an elementary automorphism :
g = en ◦ an−1 ◦ · · · ◦ a1 ◦ e1 with ai ∈ Ak̄ \ Ek̄ , ej ∈ Ek̄ \ Ak̄ .
A straightforward induction shows that we can write g as
g : (x, y) 7→ (γy d1 .d2 + · · · , δy d1 + · · · )
with γ, δ ∈ k ∗ et d1 , d2 > 1 (we wrote only the homogeneous components of higher degree).
Composing g on the left by the automorphism (x, y) 7→ (x − δγd2 y d2 , y) which is an element
of Ek we obtain an element of Aut[k 2 ] with degree strictly less than the degree of g. By
induction on the degree this ends the demonstration.
Note. While this paper was submitted for publication J. F. de Bobadilla very kindly wrote
to me to warn me he was the author, independently and simultaneously, of a proof of the Jung’s
theorem very similar to the one I have exposed here (see chapter 1 of [5]).
Acknowledgements. I had the opportunity to present this work at different stage of its
elaboration, successively at the ENS of Lyon, the Federal University of Porto Alegre and the
University of Rennes. Each times the numerous remarks and suggestions I received have been of
great benefit. I would also want to thanks also to Thierry Vust whose suggestions allowed me to
present a more readable article.
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Stéphane Lamy
Université Lyon 1
Institut Camille Jordan
21 Avenue Claude Bernard
69622 Villeurbanne Cedex
France
Current email : [email protected]
Current web page : https://www.math.univ-toulouse.fr/~slamy/
18